Theorem riotasv2s 38341

Description: The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5394) in the form of a substitution instance. Special case of riota2f 7386. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Hypothesis

Ref	Expression
riotasv2s.2	⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))

Assertion

Ref	Expression
riotasv2s	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶   𝑥,𝐸,𝑦   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝐶(𝑦)   𝐷(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem riotasv2s

Step	Hyp	Ref	Expression
1		3simpc 1147	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → (𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)))
2		simp1 1133	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐴 ∈ 𝑉)
3		riotasv2s.2	. . . . . 6 ⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
4		nfra1 3275	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)
5		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
6	4, 5	nfriota 7374	. . . . . 6 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
7	3, 6	nfcxfr 2895	. . . . 5 ⊢ Ⅎ𝑦𝐷
8	7	nfel1 2913	. . . 4 ⊢ Ⅎ𝑦 𝐷 ∈ 𝐴
9		nfv 1909	. . . . 5 ⊢ Ⅎ𝑦 𝐸 ∈ 𝐵
10		nfsbc1v 3792	. . . . 5 ⊢ Ⅎ𝑦[𝐸 / 𝑦]𝜑
11	9, 10	nfan 1894	. . . 4 ⊢ Ⅎ𝑦(𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)
12	8, 11	nfan 1894	. . 3 ⊢ Ⅎ𝑦(𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑))
13		nfcsb1v 3913	. . . 4 ⊢ Ⅎ𝑦⦋𝐸 / 𝑦⦌𝐶
14	13	a1i 11	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → Ⅎ𝑦⦋𝐸 / 𝑦⦌𝐶)
15	10	a1i 11	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → Ⅎ𝑦[𝐸 / 𝑦]𝜑)
16	3	a1i 11	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
17		sbceq1a 3783	. . . 4 ⊢ (𝑦 = 𝐸 → (𝜑 ↔ [𝐸 / 𝑦]𝜑))
18	17	adantl 481	. . 3 ⊢ (((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) ∧ 𝑦 = 𝐸) → (𝜑 ↔ [𝐸 / 𝑦]𝜑))
19		csbeq1a 3902	. . . 4 ⊢ (𝑦 = 𝐸 → 𝐶 = ⦋𝐸 / 𝑦⦌𝐶)
20	19	adantl 481	. . 3 ⊢ (((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) ∧ 𝑦 = 𝐸) → 𝐶 = ⦋𝐸 / 𝑦⦌𝐶)
21		simpl 482	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 ∈ 𝐴)
22		simprl 768	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐸 ∈ 𝐵)
23		simprr 770	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → [𝐸 / 𝑦]𝜑)
24	12, 14, 15, 16, 18, 20, 21, 22, 23	riotasv2d 38340	. 2 ⊢ (((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) ∧ 𝐴 ∈ 𝑉) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶)
25	1, 2, 24	syl2anc 583	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2877  ∀wral 3055  [wsbc 3772  ⦋csb 3888  ℩crio 7360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-riotaBAD 38336

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-riota 7361  df-undef 8259

This theorem is referenced by: (None)